2020/09/29 by Michael Voit, Voit, Michael, Jeannette H. C. Woerner +1 · 2 citations
Mathematics · #60B20 #60F05 #60F15 #60J60 #60K35 #70F10 #82C22 #Advanced Mathematical Identities #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Mathematical functions and polynomials #Probability (math.PR) #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.2009.13928
openalex publication_date 2020/09/29 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
We study Bessel and Dunkl processes (Xt,k)t\≥0 on mathbb RN with\npossibly multivariate coupling constants k\≥0. These processes describe\ninteracting particle systems of Calogero-Moser-Sutherland type with N\nparticles. For the root systems AN-1 and BN these Bessel processes are\nrelated with \β-Hermite and \β-Laguerre ensembles. Moreover, for the\nfrozen case k=\∞, these processes degenerate to deterministic or pure\njump processes. We use the generators for Bessel and Dunkl processes of types A\nand B and derive analogues of Wigner's semicircle and Marchenko-Pastur limit\nlaws for N\→\∞ for the empirical distributions of the particles with\narbitrary initial empirical distributions by using free convolutions. In\nparticular, for Dunkl processes of type B new non-symmetric semicircle-type\nlimit distributions on mathbb R appear. Our results imply that the form of\nthe limiting measures is already completely determined by the frozen processes.\nMoreover, in the frozen cases, our approach leads to a new simple proof of the\nsemicircle and Marchenko-Pastur limit laws for the empirical measures of the\nzeroes of Hermite and Laguerre polynomials respectively.\n